#8818036

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#46790	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(17,18)(19,26)(20,28)(21,24)(22,23)(25,30)(27,31)(29,32)(33,34,39,36)(35,38,43,40)(37,41,46,44)(42,45,48,47);(1,2)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,32)(29,30)(33,34,39,36)(35,40,43,38)(37,44,46,41)(42,45,48,47);(1,2,5,8)(3,9,11,6)(4,7,13,12)(10,15,16,14)(17,18,23,22)(19,24,21,26)(20,28,30,25)(27,32,29,31)(33,35,39,43)(34,40,36,38)(37,42,46,48)(41,47,44,45);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,20,23,30)(18,25,22,28)(19,27,21,29)(24,31,26,32)(33,36,39,34)(35,38,43,40)(37,44,46,41)(42,45,48,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,21,23,19)(18,24,22,26)(20,29,30,27)(25,31,28,32)(33,37)(34,41)(35,42)(36,44)(38,45)(39,46)(40,47)(43,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,22,23,18)(19,24,21,26)(20,28,30,25)(27,31,29,32)(33,38)(34,35)(36,43)(37,45)(39,40)(41,42)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8725232);